Characteristic classes of the Hilbert schemes of points on non-compact simply-connected surfaces

We prove a closed formula expressing any multiplicative characteristic class evaluated on the tangent bundle of the Hilbert schemes of points on a non-compact simply-connected surface. As a corollary, we deduce a closed formula for the Chern character of the tangent bundles of these Hilbert schemes. We also give a closed formula for the multiplicative characteristic classes of the tautological bundles associated to a line bundle on the surface. We finally remark which implications the results here have for the Hilbert schemes of points of an arbitrary surface.

💡 Research Summary

The paper addresses the problem of computing multiplicative characteristic classes of the tangent bundles of Hilbert schemes of points on a non‑compact, simply‑connected complex surface (X). While extensive results exist for compact surfaces—most notably Göttsche’s formulas for Betti numbers and Lehn’s work on tautological bundles—the non‑compact case has remained largely unexplored. The authors fill this gap by deriving a closed, universal formula that works for any multiplicative characteristic class (c_{\Phi}) evaluated on (T_{X^{